# Ig r y p o c 2 extend b to a basis for c n by building

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Unformatted text preview: T H • If x0 is a critical point such that H(x0 ) is positive deﬁnite, then f has a local minimum at x0 . • If x0 is a critical point such that H(x0 ) is negative deﬁnite (i.e., zT Hz &lt; 0 for all z = 0 or, equivalently, −H is positive deﬁnite), then f has a local maximum at x0 . IG R Exercises for section 7.6 Y P 7.6.1. Which of the following matrices are positive deﬁnite? ⎛ ⎞ ⎛ ⎞ ⎛ 1 −1 −1 20 6 8 2 A = ⎝ −1 5 1⎠. B = ⎝ 6 3 0⎠. C = ⎝0 −1 1 5 808 2 O C 0 6 2 ⎞ 2 2⎠. 4 7.6.2. Spring-Mass Vibrations. Two masses m1 and m2 are suspended between three identical springs (with spring constant k ) as shown in Figure 7.6.7. Each mass is initially displaced from its equilibrium position by a horizontal distance and released to vibrate freely (assume there is no vertical displacement). m1 m2 x1 x2 m1 m2 Figure 7.6.7 Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.6 Positive Deﬁnite Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 571 (a) If xi (t) denotes the horizontal displacement of mi from equilibrium at time t, show that Mx = Kx, where It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu M= m1 0 0 m2 , x= x1 (t) x2 (t) , and K=k 2 −1 −1 2 . (Consider a force directed to the left to be positive.) Notice that the mass-stiﬀness equation Mx = Kx is the matrix version of Hooke’s law F = kx, and K is positive deﬁnite. (b) Look for a solution of the form x = eiθt v for a constant vector v, and show that this reduces the problem to solving an algebraic equation of the form Kv = λMv (for λ = −θ2 ). This is called a generalized eigenvalue problem because when M = I we are back to the ordinary eigenvalue problem. The generalized eigenvalues λ1 and λ2 are the roots of the equation det (K − λM) = 0—ﬁnd them when k = 1, m1 = 1, and m2 = 2, and describe the two modes of vibration. (c) Take m1 = m2 = m, and...
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